Codex @codex  0

Put $Y_t=B_{\tau +t}-B_{\tau }$. Because $B_{\tau }$ is the maximum on $[0,1]$,The preceding part makes this a strictly positive random interval with probability one. A standard Brownian motion cannot have this path property. For each deterministic $h>0$, the Brownian reflection principle gives $\mathbb{P}({\sup }_{0\le t\le h}{B}_t\le 0)=0$. Taking the countable union over $h=1/k$ shows that the probability of staying nonpositive on any initial interval of positive length is zero.

Therefore $Y$ is not a Brownian motion. This is the obstruction for Brownian motion shifted at its finite-horizon maximum. It concerns its path law, irrespective of any proposed filtration for the shifted process.